Forward-Intersection Spectral Latent Dynamics

Integrate this at the transition layer of a latent world model or structured state-space model. Let an encoder produce z_t = phi_theta(x_t) in R^d, and learn a matrix A such that z_{t+1} approximately equals A z_t. Construct X and Y from a rolling buffer of consecutive latent states and estimate the transition with either the trainable A or the least-squares estimate \widehat A = YX^\dagger. Every N optimizer steps, freeze A and compute the compatible subspace. Start with Q_0 = I_d. For level j, form M = [Q_j, -A Q_j], compute an SVD M = U Sigma V^T, retain right-singular vectors associated with singular values below tolerance tau, and map their first r_j coordinates through Q_j to obtain candidate intersection vectors. More robustly under noise, compute principal angles between range(Q_j) and range(AQ_j), retain directions whose angles are below tau, and orthonormalize them with QR to obtain Q_{j+1}. Stop when r_{j+1}=r_j or after J levels. During subsequent forward passes use z_next = Q_j(Q_j^T A Q_j)Q_j^T z rather than Az. Add the usual one-step and multi-step reconstruction losses; optionally add a penalty ||(I-Q_jQ_j^T)Az_t||^2 so the learned transition respects the selected subspace. Recompute Q periodically and initially stop gradients through the SVD basis. The paper supplies the intersection construction and projected operator; tau, refresh period N, rollout horizon, and penalty weight are empirical. First test a 2-layer MLP encoder on Lorenz-63 or nonlinear pendulum trajectories with d=32, comparing j=0, 1, and 2 against the unfiltered baseline for 1-, 10-, and 100-step prediction. Success means lower long-horizon error, fewer transition eigenvalues with modulus above one, and less rollout drift at equal parameter count.